The term $\frac{\partial T}{\partial t} $ expresses the $local$ change of temperature with time. If no winds are blowing, temperature may change as well, for example due to insolation (cooler at night, warmer during the day), seasonality and so on.
Therefore, the term $\frac{DT}{Dt}$ on the left-hand side of the equation makes it maybe it more clear. It is not a component of the derivate, it is the total derivative, decomposed in local and advective terms. So advection is one of the components of $\frac{DT}{Dt}$.